CS137: Electronic Design Automation

1. CS137:Electronic Design Automation Day 6: January 23, 2002 Partitioning (Intro, KLFM)

2. Today • Partitioning • why important • practical attack • variations and issues

3. Motivation (1) • Divide-and-conquer • trivial case: decomposition • smaller problems easier to solve • net win, if super linear • 2T(n/2) +Part(n) < T(n) • problems with sparse connections or interactions • Exploit structure • limited cutsize is a common structural property • random graphs would not have as small cuts

4. Motivation (2) • Cut size (bandwidth) can determine area • Minimizing cuts • minimize interconnect requirements • increases signal locallity • Chip (board) partitioning • minimize IO • Direct basis for placement

5. N/2 N/2 Bisection Bandwidth • Partition design into two equal size halves • Minimize wires (nets) with ends in both halves • Number of wires crossing is bisection bandwidth • lower bw = more locality cutsize

6. Bisection is lower-bound on IC width Apply wire dominated (recursively) N/2 N/2 Interconnect Area

7. Classic Partitioning Problem • Given: netlist of interconnect cells • Partition into two (roughly) equal halves (A,B) • minimize the number of nets shared by halves • “Roughly Equal” • balance condition: (0.5-d)N|A|(0.5+d)N

8. Balanced Partitioning • NP-complete for general graphs • [ND17: Minimum Cut into Bounded Sets, Garey and Johnson] • Reduce SIMPLE MAX CUT • Reduce MAXIMUM 2-SAT to SMC • Unbalanced partitioning poly time • Many heuristics/attacks

9. KL FM Partitioning Heuristic • Greedy, iterative • pick cell that decreases cut and move it • repeat • small amount of: • look past moves that make locally worse • randomization

10. Fiduccia-Mattheyses(Kernighan-Lin refinement) • Start with two halves (random split?) • Repeat until no updates • Start with all cells free • Repeat until no cells free • Move cell with largest gain (balance allows) • Update costs of neighbors • Lock cell in place (record current cost) • Pick least cost point in previous sequence and use as next starting position • Repeat for different random starting points

11. Efficiency • Pick move candidate with little work • Update costs on move cheaply • Efficient data structure • update costs cheap • cheap to find next move

12. Ordering and Cheap Update • Keep track of Net gain on node == delta net crossings to move a node • cut cost after move = cost - gain • Calculate node gain as sigma net gains for all nets at that node • Sort by gain

13. FM Cell Gains Gain = Delta in number of nets crossing between partitions

14. FM Cell Gains Gain = Delta in number of nets crossing between partitions 0 -4 1 +4 0 2

15. After move node? • Update cost each • Newcost=cost-gain • Also need to update gains • on all nets attached to moved node • roll up to all nodes affected by those nets

16. FM Recompute Cell Gain • For each net, keep track of number of cells in each partition [F(net), T(net)] • Move update:(for each net on moved cell) • if T(net)==0, increment gain on F side of net • (think -1 => 0) • if T(net)==1, decrement gain on T side of net • (think 1=>0) • decrement F(net), increment T(net) • if F(net)==1, increment gain on F cell • if F(net)==0, decrement gain on all cells (T)

17. FM Recompute (example)

18. FM Recompute (example)

19. Partition Counts A,B Two gain arrays One per partition Key: constant time cell update Cells successors (consumers) inputs locked status FM Data Structures

20. FM Optimization Sequence (ex)

21. FM Running Time? • Randomly partition into two halves • Repeat until no updates • Start with all cells free • Repeat until no cells free • Move cell with largest gain • Update costs of neighbors • Lock cell in place (record current cost) • Pick least cost point in previous sequence and use as next starting position • Repeat for different random starting points

22. FM Running Time • Claim: small number of passes (constant?) to converge • Small (constant?) number of random starts • N cell updates each round (swap) • Updates K + fanout work (avg. fanout K) • assume K-LUTs • Maintain ordered list O(1) per move • every io move up/down by 1 • Running time: O(KN) • Algorithm significant for its speed (more than quality)

23. FM Starts? So, FM gives a not bad solution quickly 21K random starts, 3K network -- Alpert/Kahng

24. Weaknesses? • Local, incremental moves only • hard to move clusters • no lookahead • [see example] • Looks only at local structure

25. Improving FM • Clustering • technology mapping • initial partitions • runs • partition size freedom • replication Following comparisons from Hauck and Boriello ‘96

26. Clustering • Group together several leaf cells into cluster • Run partition on clusters • Uncluster (keep partitions) • iteratively • Run partition again • using prior result as starting point • instead of random start

27. Clustering Benefits • Catch local connectivity which FM might miss • moving one element at a time, hard to see move whole connected groups across partition • Faster (smaller N) • METIS -- fastest research partitioners exploits heavily • FM work better w/ larger nodes (???)

28. How Cluster? • Random • cheap, some benefits for speed • Greedy “connectivity” • examine in random order • cluster to most highly connected • 30% better cut, 16% faster than random • Spectral (next time) • look for clusters in placement • (ratio-cut like) • Brute-force connectivity (can be O(N2))

29. LUT Mapped? • Better to partition before LUT mapping.

30. Initial Partitions? • Random • Pick Random node for one side • start imbalanced • run FM from there • Pick random node and Breadth-first search to fill one half • Pick random node and Depth-first search to fill half • Start with Spectral partition

31. Initial Partitions • If run several times • pure random tends to win out • more freedom / variety of starts • more variation from run to run • others trapped in local minima

32. Number of Runs

33. Number of Runs • 2 - 10% • 10 - 18% • 20 <20% (2% better than 10) • 50 (4% better than 10) • …but?

34. FM Starts? 21K random starts, 3K network -- Alpert/Kahng

35. Unbalanced Cuts • Increasing slack in partitions • may allow lower cut size

36. Unbalanced Partitions Following comparisons from Hauck and Boriello ‘96

37. Replication • Trade some additional logic area for smaller cut size • Net win if wire dominated Replication data from: Enos, Hauck, Sarrafzadeh ‘97

38. Replication • 5% => 38% cut size reduction • 50% => 50+% cut size reduction

39. What Bisection doesn’t tell us • Bisection bandwidth purely geometrical • No constraint for delay • I.e. a partition may leave critical path weaving between halves

40. Critical Path and Bisection Minimum cut may cross critical path multiple times. Minimizing long wires in critical path => increase cut size.

41. So... • Minimizing bisection • good for area • oblivious to delay/critical path

42. Partitioning Summary • Decompose problem • Find locality • NP-complete problem • linear heuristic (KLFM) • many ways to tweak • Hauck/Boriello, Karypis • even better with replication • only address cut size, not critical path delay

43. Reading and References • Day 8 – pickup off web • Day 9 – hardcopy today • May be an additional • Day 10 – pickup off web • Today’s supplemental references linked to reading • (Hauck, Karypis, Alpert…)

44. Today’s Big Ideas: • Divide-and-Conquer • Exploit Structure • Look for sparsity/locality of interaction • Techniques: • greedy • incremental improvement • randomness avoid bad cases, local minima • incremental cost updates (time cost) • efficient data structures